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Beta laws: arcsine, uniform, semicircle

The Beta law on \( {[0,1]} \) with parameters \( {a>0} \) and \( {b>0} \) has density

\[ x\mapsto \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} x^{a-1}(1-x)^{b-1}\mathbf{1}_{[0,1]}(x). \]

It is sometimes more convenient to work on the interval \( {[-1,1]} \) instead of \( {[0,1]} \). Indeed, with the substitution \( {y=2x-1} \), we get the Beta density on \( {[-1,1]} \) :

\[ y\mapsto \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}2^{1-a-b} (1+y)^{a-1}(1-y)^{b-1}\mathbf{1}_{[-1,1]}(y). \]

In particular, when \( {a=b} \) then it boils down to

\[ y\mapsto \frac{\Gamma(2a)}{\Gamma(a)^2}2^{1-2a} (1-y^2)^{2(a-1)}\mathbf{1}_{[-1,1]}(y). \]

For \( {a=3/4} \) we get the arcsine law, for \( {a=1} \) the uniform law, and for \( {a=5/4} \) the semicircle law. The Jacobi polynomials are orthogonal for this \( {(a,b)} \)-beta law on \( {[-1,1]} \). We recover the Chebyshev polynomials of the first and second kind in the arcsine and semicircle cases respectively, and the Legendre polynomials in the uniform case.

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